# Binary cyclic group

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In mathematics, the binary cyclic group of the n-gon is the cyclic group of order 2n, $C_{2n}$, thought of as an extension of the cyclic group $C_n$ by a cyclic group of order 2. It is the binary polyhedral group corresponding to the cyclic group.[1]

In terms of binary polyhedral groups, the binary cyclic group is the preimage of the cyclic group of rotations ($C_n < \operatorname{SO}(3)$) under the 2:1 covering homomorphism

$\operatorname{Spin}(3) \to \operatorname{SO}(3)\,$

of the special orthogonal group by the spin group.

As a subgroup of the spin group, the binary cyclic group can be described concretely as a discrete subgroup of the unit quaternions, under the isomorphism $\operatorname{Spin}(3) \cong \operatorname{Sp}(1)$ where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)

## References

1. ^ Coxeter, H. S. M. (1959), "Symmetrical definitions for the binary polyhedral groups", Proc. Sympos. Pure Math., Vol. 1, Providence, R.I.: American Mathematical Society, pp. 64–87, MR 0116055.